On the six-vertex model's free energy

TL;DR

This paper rigorously analyzes the six-vertex model's free energy by proving Bethe root condensation, providing explicit calculations, and exploring asymptotic behaviors related to phase transitions and model invariances.

Contribution

It offers new proofs of Bethe root existence and condensation, and computes the free energy and asymptotics of the six-vertex model with rigorous methods.

Findings

Proof of Bethe root condensation for arbitrary densities

Explicit computation of the free energy on the torus

Asymptotic expansion of partition functions near half density

Abstract

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $\Delta<1$. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches $1/2$. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $a=b=1$ and $c\ge1$, and the rotational invariance of the six-vertex model and the Fortuin-Kasteleyn percolation.